Copyright	(c) Edward Kmett 2010-2015
License	BSD3
Maintainer	ekmett@gmail.com
Stability	experimental
Portability	GHC only
Safe Haskell	None
Language	Haskell2010

Numeric.AD.Internal.Sparse

Description

Unsafe and often partial combinators intended for internal usage.

Handle with care.

Synopsis

newtype Monomial = Monomial (IntMap Int)
emptyMonomial :: Monomial
addToMonomial :: Int -> Monomial -> Monomial
indices :: Monomial -> [Int]
data Sparse a
- = Sparse !a (IntMap (Sparse a))
- | Zero
apply :: (Traversable f, Num a) => (f (Sparse a) -> b) -> f a -> b
vars :: (Traversable f, Num a) => f a -> f (Sparse a)
d :: (Traversable f, Num a) => f b -> Sparse a -> f a
d' :: (Traversable f, Num a) => f a -> Sparse a -> (a, f a)
ds :: (Traversable f, Num a) => f b -> Sparse a -> Cofree f a
skeleton :: Traversable f => f a -> f Int
spartial :: Num a => [Int] -> Sparse a -> Maybe a
partial :: Num a => [Int] -> Sparse a -> a
vgrad :: Grad i o o' a => i -> o
vgrad' :: Grad i o o' a => i -> o'
vgrads :: Grads i o a => i -> o
class Num a => Grad i o o' a | i -> a o o', o -> a i o', o' -> a i o where
- pack :: i -> [Sparse a] -> Sparse a
- unpack :: ([a] -> [a]) -> o
- unpack' :: ([a] -> (a, [a])) -> o'
class Num a => Grads i o a | i -> a o, o -> a i where
- packs :: i -> [Sparse a] -> Sparse a
- unpacks :: ([a] -> Cofree [] a) -> o
terms :: Monomial -> [(Integer, Monomial, Monomial)]
primal :: Num a => Sparse a -> a

Documentation

newtype Monomial Source #

Constructors

Monomial (IntMap Int)

emptyMonomial :: Monomial Source #

addToMonomial :: Int -> Monomial -> Monomial Source #

indices :: Monomial -> [Int] Source #

data Sparse a Source #

We only store partials in sorted order, so the map contained in a partial will only contain partials with equal or greater keys to that of the map in which it was found. This should be key for efficiently computing sparse hessians. there are only n + k - 1 choose k distinct nth partial derivatives of a function with k inputs.

Constructors

Sparse !a (IntMap (Sparse a))
Zero

Instances

(Num a, Bounded a) => Bounded (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods minBound :: Sparse a # maxBound :: Sparse a #
(Num a, Enum a) => Enum (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods succ :: Sparse a -> Sparse a # pred :: Sparse a -> Sparse a # toEnum :: Int -> Sparse a # fromEnum :: Sparse a -> Int # enumFrom :: Sparse a -> [Sparse a] # enumFromThen :: Sparse a -> Sparse a -> [Sparse a] # enumFromTo :: Sparse a -> Sparse a -> [Sparse a] # enumFromThenTo :: Sparse a -> Sparse a -> Sparse a -> [Sparse a] #
(Num a, Eq a) => Eq (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods (==) :: Sparse a -> Sparse a -> Bool # (/=) :: Sparse a -> Sparse a -> Bool #
Floating a => Floating (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods pi :: Sparse a # exp :: Sparse a -> Sparse a # log :: Sparse a -> Sparse a # sqrt :: Sparse a -> Sparse a # (**) :: Sparse a -> Sparse a -> Sparse a # logBase :: Sparse a -> Sparse a -> Sparse a # sin :: Sparse a -> Sparse a # cos :: Sparse a -> Sparse a # tan :: Sparse a -> Sparse a # asin :: Sparse a -> Sparse a # acos :: Sparse a -> Sparse a # atan :: Sparse a -> Sparse a # sinh :: Sparse a -> Sparse a # cosh :: Sparse a -> Sparse a # tanh :: Sparse a -> Sparse a # asinh :: Sparse a -> Sparse a # acosh :: Sparse a -> Sparse a # atanh :: Sparse a -> Sparse a # log1p :: Sparse a -> Sparse a # expm1 :: Sparse a -> Sparse a # log1pexp :: Sparse a -> Sparse a # log1mexp :: Sparse a -> Sparse a #
Fractional a => Fractional (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods (/) :: Sparse a -> Sparse a -> Sparse a # recip :: Sparse a -> Sparse a # fromRational :: Rational -> Sparse a #
Data a => Data (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Sparse a -> c (Sparse a) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Sparse a) # toConstr :: Sparse a -> Constr # dataTypeOf :: Sparse a -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Sparse a)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Sparse a)) # gmapT :: (forall b. Data b => b -> b) -> Sparse a -> Sparse a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Sparse a -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Sparse a -> r # gmapQ :: (forall d. Data d => d -> u) -> Sparse a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Sparse a -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Sparse a -> m (Sparse a) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Sparse a -> m (Sparse a) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Sparse a -> m (Sparse a) #
Num a => Num (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods (+) :: Sparse a -> Sparse a -> Sparse a # (-) :: Sparse a -> Sparse a -> Sparse a # (*) :: Sparse a -> Sparse a -> Sparse a # negate :: Sparse a -> Sparse a # abs :: Sparse a -> Sparse a # signum :: Sparse a -> Sparse a # fromInteger :: Integer -> Sparse a #
(Num a, Ord a) => Ord (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods compare :: Sparse a -> Sparse a -> Ordering # (<) :: Sparse a -> Sparse a -> Bool # (<=) :: Sparse a -> Sparse a -> Bool # (>) :: Sparse a -> Sparse a -> Bool # (>=) :: Sparse a -> Sparse a -> Bool # max :: Sparse a -> Sparse a -> Sparse a # min :: Sparse a -> Sparse a -> Sparse a #
Real a => Real (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods toRational :: Sparse a -> Rational #
RealFloat a => RealFloat (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods floatRadix :: Sparse a -> Integer # floatDigits :: Sparse a -> Int # floatRange :: Sparse a -> (Int, Int) # decodeFloat :: Sparse a -> (Integer, Int) # encodeFloat :: Integer -> Int -> Sparse a # exponent :: Sparse a -> Int # significand :: Sparse a -> Sparse a # scaleFloat :: Int -> Sparse a -> Sparse a # isNaN :: Sparse a -> Bool # isInfinite :: Sparse a -> Bool # isDenormalized :: Sparse a -> Bool # isNegativeZero :: Sparse a -> Bool # isIEEE :: Sparse a -> Bool # atan2 :: Sparse a -> Sparse a -> Sparse a #
RealFrac a => RealFrac (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods properFraction :: Integral b => Sparse a -> (b, Sparse a) # truncate :: Integral b => Sparse a -> b # round :: Integral b => Sparse a -> b # ceiling :: Integral b => Sparse a -> b # floor :: Integral b => Sparse a -> b #
Show a => Show (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods showsPrec :: Int -> Sparse a -> ShowS # show :: Sparse a -> String # showList :: [Sparse a] -> ShowS #
Erf a => Erf (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods erf :: Sparse a -> Sparse a # erfc :: Sparse a -> Sparse a # erfcx :: Sparse a -> Sparse a # normcdf :: Sparse a -> Sparse a #
InvErf a => InvErf (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods inverf :: Sparse a -> Sparse a # inverfc :: Sparse a -> Sparse a # invnormcdf :: Sparse a -> Sparse a #
Num a => Mode (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Associated Types type Scalar (Sparse a) :: Type Source # Methods isKnownConstant :: Sparse a -> Bool Source # isKnownZero :: Sparse a -> Bool Source # auto :: Scalar (Sparse a) -> Sparse a Source # (^) :: Scalar (Sparse a) -> Sparse a -> Sparse a Source # (^) :: Sparse a -> Scalar (Sparse a) -> Sparse a Source # (^/) :: Sparse a -> Scalar (Sparse a) -> Sparse a Source # zero :: Sparse a Source #
Num a => Jacobian (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Associated Types type D (Sparse a) :: Type Source # Methods unary :: (Scalar (Sparse a) -> Scalar (Sparse a)) -> D (Sparse a) -> Sparse a -> Sparse a Source # lift1 :: (Scalar (Sparse a) -> Scalar (Sparse a)) -> (D (Sparse a) -> D (Sparse a)) -> Sparse a -> Sparse a Source # lift1_ :: (Scalar (Sparse a) -> Scalar (Sparse a)) -> (D (Sparse a) -> D (Sparse a) -> D (Sparse a)) -> Sparse a -> Sparse a Source # binary :: (Scalar (Sparse a) -> Scalar (Sparse a) -> Scalar (Sparse a)) -> D (Sparse a) -> D (Sparse a) -> Sparse a -> Sparse a -> Sparse a Source # lift2 :: (Scalar (Sparse a) -> Scalar (Sparse a) -> Scalar (Sparse a)) -> (D (Sparse a) -> D (Sparse a) -> (D (Sparse a), D (Sparse a))) -> Sparse a -> Sparse a -> Sparse a Source # lift2_ :: (Scalar (Sparse a) -> Scalar (Sparse a) -> Scalar (Sparse a)) -> (D (Sparse a) -> D (Sparse a) -> D (Sparse a) -> (D (Sparse a), D (Sparse a))) -> Sparse a -> Sparse a -> Sparse a Source #
Num a => Grad (Sparse a) [a] (a, [a]) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods pack :: Sparse a -> [Sparse a] -> Sparse a Source # unpack :: ([a] -> [a]) -> [a] Source # unpack' :: ([a] -> (a, [a])) -> (a, [a]) Source #
Num a => Grads (Sparse a) (Cofree [] a) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods packs :: Sparse a -> [Sparse a] -> Sparse a Source # unpacks :: ([a] -> Cofree [] a) -> Cofree [] a Source #
Grads i o a => Grads (Sparse a -> i) (a -> o) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods packs :: (Sparse a -> i) -> [Sparse a] -> Sparse a Source # unpacks :: ([a] -> Cofree [] a) -> a -> o Source #
Grad i o o' a => Grad (Sparse a -> i) (a -> o) (a -> o') a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods pack :: (Sparse a -> i) -> [Sparse a] -> Sparse a Source # unpack :: ([a] -> [a]) -> a -> o Source # unpack' :: ([a] -> (a, [a])) -> a -> o' Source #
type Scalar (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse type Scalar (Sparse a) = a
type D (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse type D (Sparse a) = Sparse a

apply :: (Traversable f, Num a) => (f (Sparse a) -> b) -> f a -> b Source #

vars :: (Traversable f, Num a) => f a -> f (Sparse a) Source #

d :: (Traversable f, Num a) => f b -> Sparse a -> f a Source #

d' :: (Traversable f, Num a) => f a -> Sparse a -> (a, f a) Source #

ds :: (Traversable f, Num a) => f b -> Sparse a -> Cofree f a Source #

skeleton :: Traversable f => f a -> f Int Source #

spartial :: Num a => [Int] -> Sparse a -> Maybe a Source #

partial :: Num a => [Int] -> Sparse a -> a Source #

vgrad :: Grad i o o' a => i -> o Source #

vgrad' :: Grad i o o' a => i -> o' Source #

vgrads :: Grads i o a => i -> o Source #

class Num a => Grad i o o' a | i -> a o o', o -> a i o', o' -> a i o where Source #

Methods

pack :: i -> [Sparse a] -> Sparse a Source #

unpack :: ([a] -> [a]) -> o Source #

unpack' :: ([a] -> (a, [a])) -> o' Source #

Instances

Num a => Grad (Sparse a) [a] (a, [a]) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods pack :: Sparse a -> [Sparse a] -> Sparse a Source # unpack :: ([a] -> [a]) -> [a] Source # unpack' :: ([a] -> (a, [a])) -> (a, [a]) Source #
Grad i o o' a => Grad (Sparse a -> i) (a -> o) (a -> o') a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods pack :: (Sparse a -> i) -> [Sparse a] -> Sparse a Source # unpack :: ([a] -> [a]) -> a -> o Source # unpack' :: ([a] -> (a, [a])) -> a -> o' Source #

class Num a => Grads i o a | i -> a o, o -> a i where Source #

Methods

packs :: i -> [Sparse a] -> Sparse a Source #

unpacks :: ([a] -> Cofree [] a) -> o Source #

Instances

Num a => Grads (Sparse a) (Cofree [] a) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods packs :: Sparse a -> [Sparse a] -> Sparse a Source # unpacks :: ([a] -> Cofree [] a) -> Cofree [] a Source #
Grads i o a => Grads (Sparse a -> i) (a -> o) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods packs :: (Sparse a -> i) -> [Sparse a] -> Sparse a Source # unpacks :: ([a] -> Cofree [] a) -> a -> o Source #

terms :: Monomial -> [(Integer, Monomial, Monomial)] Source #

The value of the derivative of (f*g) of order mi is

sum [a * primal (partialS (indices b) f) * primal (partialS (indices c) g) | (a,b,c) <- terms mi ]

It is a bit more complicated in mul below, since we build the whole tree of derivatives and want to prune the tree with Zeros as much as possible. The number of terms in the sum for order mi as of differentiation has sum (map (+1) as) terms, so this is *much* more efficient than the naive recursive differentiation with 2^sum as terms. The coefficients a, which collect equivalent derivatives, are suitable products of binomial coefficients.

primal :: Num a => Sparse a -> a Source #